Sequences and S...

Let x, y, z be three positive prime numbers. The progression in which $$\sqrt{x}$$, $$\sqrt{y}$$, $$\sqrt{z}$$ can be three terms (not necessarily consecutive) is

Let $$\displaystyle a_{1},a_{2},a_{3},....,a_{11}$$ be real numbers satisfying $$\displaystyle a_{1}=15,27-2a_{2}> 0\:$$ and $$\: \: a_{k}=2a_{k-1}-a_{k-2}$$ for $$k = 3, 4, ......., 11$$.

It is known that $$\sum_{r=1}^{\infty }\frac{1}{\left ( 2r-1 \right )^{2}}=\frac{\pi ^{2}}{8}$$.  Then $$\sum_{r=1}^{\infty }\frac{1}{r^{2}}$$ is equal to

Evaluate $$\sum_{r=1}^{n}\left [ \sum_{k=1}^{r}k \right ] \left [ \log_{1/2}\sqrt{(4x-x^{2})} \right ]^{r}$$. Find $$x$$ for which summation is a finite number as $$n\rightarrow \infty $$

if $$\displaystyle \frac{1}{1^2}\, +\, \displaystyle \frac{1}{2^2}\, +\, \displaystyle \frac{1}{3^2}$$ + .......... upto $$\infty\, =\, \displaystyle \frac{\pi^2}{6}$$, then $$\displaystyle \frac{1}{1^2}\, +\, \displaystyle \frac{1}{3^2}\, +\, \displaystyle \frac{1}{5^2}$$ + ........... = .......... .

Let $$\displaystyle \left \{ a_{n} \right \}\: and\: \left \{ b_{n} \right \}$$ are two sequences given by $$\displaystyle a_{n}=\left ( x \right )^{1/2^{n}}+\left ( y \right )^{1/2^{n}}\: \: and\: \: b_{n}=\left ( x \right )^{1/2^{n}}-\left ( y \right )^{1/2^{n}}$$ for all n $$\displaystyle \epsilon $$ N. The value of $$\displaystyle a_{1}a_{2}a_{3}...a_{n}$$ is equal to

If a number sequence begins $$1, 3, 4, 6, 7, 9, 10, 12 . . .$$, which of the following numbers does NOT appear in the sequence?

Sum of the series $$\displaystyle \sum_{r=1}^{88}\left ( -1 \right )^{r+1}\frac{1}{\sin ^{2}\left ( r+1 \right )^{\circ}-\sin ^{2}1^{\circ}}$$ is equal to

The expression
$$\displaystyle \frac {2^2 + 1} {2^2 - 1} + \frac {3^2 + 1} {3^2 - 1} + \frac {4^2 + 1} {4^2 - 1} + ........... + \frac {(2011)^2 + 1} {(2011)^2 - 1} $$
lies in the interval

The value of $$1000\left[\dfrac {1}{1\times 2}+\dfrac {1}{2\times 3}+\dfrac {1}{3\times 4}+...+\dfrac {1}{999\times 1000}\right]$$ is equal to